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Polynomial Functions and Their Graphs. Definition of a Polynomial Function. Let n be a nonnegative integer and let a n , a n -1 ,…, a 2 , a 1 , a 0 , be real numbers with a n ¹ 0. The function defined by f ( x ) = a n x n + a n-1 x n -1 + … + a 2 x 2 + a 1 x + a 0 - PowerPoint PPT Presentation
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Polynomial Functions and Their
Graphs
Definition of a Polynomial Function
Let n be a nonnegative integer and let an, an-
1,…, a2, a1, a0, be real numbers with an 0. The function defined by
f (x) anxn an-1x
n-1 … a2x2 a1x a0
is called a polynomial function of x of degree n. The number an, the coefficient of the variable to the highest power, is called the leading coefficient.
Smooth, Continuous Graphs
Two important features of the graphs of polynomial functions are that they are smooth and continuous. By smooth, we mean that the graph contains only rounded curves with nosharp corners. By continuous, we mean that the graph has no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. These ideas are illustrated in the figure.
Smooth rounded corner
Smooth rounded corner
Smooth rounded corner
Smooth rounded corner
x x
y y
The Leading Coefficient Test
As x increases or decreases without bound, the graph of the polynomial function
f (x) anxn an-1x
n-1 an-2xn-2… a1x a0 (an 0)
eventually rises or falls. In particular,
1. For n odd: an 0 an 0
As x increases or decreases without bound, the graph of the polynomial function
f (x) anxn an-1x
n-1 an-2xn-2… a1x a0 (an 0)
eventually rises or falls. In particular,
1. For n odd: an 0 an 0
If the leading coefficient is positive, the graph falls to the left and rises to the right.
If the leading coefficient is negative, the graph rises to the left and falls to the right.
Rises right
Falls left
Falls right
Rises left
As x increases or decreases without bound, the graph of the polynomial function
f (x) anxn an-1x
n-1 an-2xn-2… a1x a0 (an 0)
eventually rises or falls. In particular,
1. For n even: an 0 an 0
As x increases or decreases without bound, the graph of the polynomial function
f (x) anxn an-1x
n-1 an-2xn-2… a1x a0 (an 0)
eventually rises or falls. In particular,
1. For n even: an 0 an 0
If the leading coefficient is positive, the graph rises to the left and to the right.
If the leading coefficient is negative, the graph falls to the left and to the right.
Rises right Rises left
Falls left
Falls right
The Leading Coefficient Test
Text Example
Solution Because the degree is odd (n 3) and the leading coefficient, 1, is positive, the graph falls to the left and rises to the right, as shown in the figure.
Use the Leading Coefficient Test to determine the end behavior of the graph of Graph the quadratic function f (x) x3 3x2 x 3.
Falls left
yRises right
x
x4 4x3 4x2 We now have a polynomial equation.
Solution We find the zeros of f by setting f (x) equal to 0.
x2(x2 4x 4) 0 Factor out x2.
x2(x 2)2 0 Factor completely.
x2 or (x 2)2 0 Set each factor equal to zero. x 0 x 2 Solve for x.
x4 4x3 4x2 0 Multiply both sides by 1. (optional step)
Find all zeros of f (x) x4 4x3 4x2.
Text Example
Multiplicity and x-Intercepts
If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.
Example
• Find the x-intercepts and multiplicity of f(x) = 2(x+2)2(x-3)
Solution:
• x=-2 is a zero of multiplicity 2 or even
• x=3 is a zero of multiplicity 1 or odd
Graphing a Polynomial Function
f (x) anxn an-1x
n-1 an-2xn-2 a1x a0 (an 0)
1. Use the Leading Coefficient Test to determine the graph's end behavior.
2. Find x-intercepts by setting f (x) 0 and solving the resulting polynomial equation. If there is an x-intercept at r as a result of (x r)k in the complete factorization of f (x), then:a. If k is even, the graph touches the x-axis at r and turns around.
b. If k is odd, the graph crosses the x-axis at r.
c. If k > 1, the graph flattens out at (r, 0).
3. Find the y-intercept by setting x equal to 0 and computing f (0).
Step 1 Determine end behavior. Because the degree is even (n 4) and the leading coefficient, 1, is positive, the graph rises to the left and the right:
Solution
Graph: f (x) x4 2x2 .
y
x
Rises left
Rises right
Text Example
x2 1)x2 1) Factor.
(x 1)2(x 1)2 0 Express the factoring in more compact notation. (x 1)2 or (x 1)2 0 Set each factor
equal to zero. x 1 x 1 Solve for x.
(x 1)(x 1)(x 1)(x 1) 0 Factor completely.
Step 2 Find the x-intercepts (zeros of the function) by setting f (x)
x4 2x2 0
Solution
Graph: f (x) x4 2x2 .
Text Example cont.
Step 2 We see that -1 and 1 are both repeated zeros with multiplicity 2. Because of the even multiplicity, the graph touches the x-axis at 1 and 1 and turns around. Furthermore, the graph tends to flatten out at these zeros with multiplicity greater than one:
Solution
Graph: f (x) x4 2x2 .
Rises left
Rises right
x
y
1 1
Text Example cont.
f 04 2 • 02 1
Step 3 Find the y-intercept. Replace x with 0 in f (x) x 4x 1.
There is a y-intercept at 1, so the graph passes through (0, 1).
Solution
Graph: f (x) x4 2x2 .
1Rises left
Rises right
x
y
1 1
Text Example cont.
Solution
Graph: f (x) x4 2x2 .
y
x
Text Example cont.
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